An Extension of the Pizzetti Formula for Polyharmonic Functions

We represent the integral over the unit ball B in Rⁿ of any poly-harmonic function u(x) of degree m as a linear combination with constant coefficients of the integrals of its Laplacians ^ju (j = 0,...,m - 1) over any fixed(n - 1)-dimensional hypersphere S() of radius (0 1). In case = 0 theformula reduces to the classical Pizzetti formula. In particular, the cubature formula derived here integrates exactly all algebraic polynomials of degree 2m - 1.